The Sally Clark BN (Fenton)

The Sally Clark BN (Fenton)

Sensitivity analysis?

Precise probabilism

Run with a single probability measure

Imprecise probabilism

Run with various measures, especially focusing on extreme plausible values

The worries

  • No clear principles for choosing ranges

  • No weighting of options

Higher-order and honesty

Precise estimate

This type of fiber occurs in 25% of carpets in this town…

Estimate with uncertainty

…based on a random sample of 40/100/200 carpets.

Higher-order and honesty

SC with HOP

SC with HOP

SC with HOP: Likelihood ratios

Weight of evidence

Basic intuitions

  • Items of evidence leading to different expected values should be able to have the same weight

  • Items of evidence leading to the same value should be able to have different weights

  • In simple set up, such as Bernoulli trials, weight should increase with the number of observations

  • For unimodal distributions, the wider the distribution associated with a given piece of evidence, the less weight this evidence has

Modular approach

  • weights as associated with distributions

  • weight of evidence results from distribution weight comparison

Shannon information

  • The right path is: 011.

  • There are \(m=8\) possible destinations by decisions at \(\log_2(8)=3\) forks

  • Initially you thought the probability that it is the right one was .5

  • Now you know it is the right one. Surprise: \(\frac{1}{.5}=2\)

  • One bit of information: \(\log_2\left(\frac{1}{.5}\right)=1\)

  • Complete instruction: \(\log_2\left(\frac{1}{.5^3}\right)=3\)

  • Notice that \(\log_2\left(\frac{1}{a}\right)= - \log_2(a)\), so: \(h(x) = - \log_2 \mathsf{P(x)}\)

Entropy: average Shannon information

\(H(X) = \sum \mathsf{P}(x_i) \log_2 \frac{1}{\mathsf{P}(x_i)} = - \sum \mathsf{P}(x_i) \log_2 \mathsf{P}(x_i)\)

Conceptualization

The expected amount of information you receive once you learn the value of \(X\).

Weight of evidence: explication

Absolute distribution weight (adw)

The more informative a piece of evidence is, as compared to the uniform distribution, the more weight it has, on scale 0 to 1.

\(\mathsf{adw(posterior)} = 1 - \left( \frac{H(\mathsf{posterior})}{H(\mathsf{uniform})}\right)\)

Relative distribution weight (rdw)

\(\mathsf{rdw(posterior, prior)} = 1 - \left( \frac{H(\mathsf{posterior})}{H(\mathsf{prior})}\right)\)

Weight of evidence (wDelta)

\(\mathsf{wDelta}(\mathsf{posterior, prior}) = \vert \mathsf{adw}(\mathsf{posterior}) - \mathsf{adw}(\mathsf{prior})\vert\)

Weights of beta distributions

Weights of beta distributions

Weights of beta distributions

Weights of beta distributions

Weight in SC with HOP